Singular Value Inequalities for Hilbert Space Operators

Filomat 32:8 (2018), 2861–2866 Published by Faculty of Sciences and Mathematics, https://doi.org/10.2298/FIL1808861T University of Nis,ˇ Serbia Available at: http://www.pmf.ni.ac.rs/filomat

Singular Value inequalities for Hilbert space Operators

Mahdi Taleb Alfakhra, Mohsen Erfanian Omidvara

aDepartment of Mathematics, Mashhad Branch, Islamic Azad University, Mashhad, Iran.

Abstract. In this paper we show that if Ai, Bi, Xi are Hilbert space operators such that Xi is compact i = 1, 2,..., n and f , 1 are non-negative continuous functions on [0, ) with f (t)1(t) = t for all t [0, ), also ∞ ∈ ∞ h is non-negative increasing operator convex function on [0, ), then ∞      Xn Xn Xn     2 2  h sj  ωiAi∗Xi∗Bi sj  ωih(Ai∗ f ( Xi∗ ) Ai) ωih(Bi∗1( Xi ) Bi)    ≤  | | ⊕ | |  i=1 i=1 i=1 Pn for j = 1, 2,... and i=1 ωi = 1. Also, applications of some inequalities are given.

1. Introduction

Let be a complex Hilbert space with inner product , and let ( ) denote the C∗-algebra of all boundedH linear operators on a complex separable Hilbert spaceh· ·i . ForB a compactH operator X ( ), let H 1 ∈ B H s (X) > s (X) > denote the singular values of X; i.e., the eigenvalues of X = (X∗X) 2 (The absolute value 1 2 ··· | | of X), arranged in decreasing order are repeated according to multiplicity. Note that sj(X) = sj(X∗) = sj( X ) for j = 1, 2,.... For A, B ( ), we utilize the direct sum notation A B for the block-diagonal operator| | " # ∈ B H ⊕ A 0 deﬁned on . It has been shown in [8] that if X and Y are compact operators, then 0 B H ⊕ H

sj(X + Y) 6 2sj(X Y) ⊕ for j = 1, 2,.... The usual operator norm of an operator A ( ) is denote by A = sup Ax : x = 1 . As an immediate consequence of the min-max principle∈ (see B H e.g.,[2, p. 75]), if||A,||B, and X{||are|| in ||( || ) such} that X is compact, then B H

sj(AXB) 6 A B sj(X) (1) || || || || 1 p p for j = 1, 2,.... For 1 6 p < , the Schatten p-norm of compact operator A is deﬁned by A p = (tr A ) , where tr is the usual trace functional.∞ One can show that || || | | A B = max( A , B ) || ⊕ || || || || ||

2010 Mathematics Subject Classification. Primary 47A30; Secondary 15A18, 47A63, 47TB10. Keywords. singular value, unitarily Invariant norm, positive operator Received: 21 June 2017; Accepted: 26 December 2017 Communicated by Fuad Kittaneh Email addresses: [email protected] (Mahdi Taleb Alfakhr), [email protected] (Mohsen Erfanian Omidvar) M. Taleb Alfakhr, M. E. Omidvar / Filomat 32:8 (2018), 2861–2866 2862 and p p 1 A B = ( A + B ) p . || ⊕ ||p || ||p || ||p In addition to the usual operator norm, which is defined on all of ( ), we consider unitarily invariant norms . Each of these norms is defined on a norm ideal containedB inH the ideal of compact operators, and for the||| sake·||| of brevity, we will make no explicit mention of this ideal. Thus, when we consider X , we are assuming that the operator X belongs to the norm ideal associated with . Moreover, each||| unitarily||| invariant norm is symmetric gauge function of the singular values, and|||· is||| characterized by equality X = UXV for|||·||| all operator X and for all unitary operators U and V ( ). For the general theory of |||unitarily||| ||| invariant||| norms, we refer to [2] or [7]. It has been shown by Bhatia∈ B andH Kittaneh in [4] that if A, B are compact operators in ( ), then B H 2 A∗B 6 AA∗ + BB∗ (2) ||| ||| ||| ||| and

A∗B + B∗A 6 AA∗ + BB∗ (3) ||| ||| ||| ||| for every unitarily invariant norm. The inequality (2) has attracted the attention of several mathematicians, and diﬀerent proof of a stronger version of it have been given. See [3], [8], [9] and [11]. It has been shown by Kittaneh [10] the generalized from of the mixed Schwarz inequality, that if A is an operator in ( ) and f and 1 are non-negative functions on [0, ) which are continuous and satisfy the relation f (t)1B(t)H= t for all t [0, ), then ∞ ∈ ∞

Ax, y f ( A∗ )x 1( A )y |h i| ≤ | | | | for all x and y in . In this paper, weH generalize inequalities (2) and (3) and present a bound that involves operator A and B.

2. Main Results In this section, we establish generalized singular value inequalities for Hilbert space operator. The following Lemmas are essential role in our analysis. The ﬁrst one on the Mixed schwarz inequality has been proven by Kittaneh [10]. " # AC Lemma 2.1. Let A, B, C ( ), such that A and B are positive, then T = ∗ is a positive in B( ) if ∈ B H CB H ⊕ H and only if Cx, y 2 6 Ax, x By, y for all x, y in . |h i| h ih i H We prove the second one by Lemma 2.1. Lemma 2.2. Let A, B and X be operators in ( ) and let f and 1 be non-negative continuous functions on [0, ) that satisfy the relation f (t)1(t) = t for all t B[0,H ), then ∞ ∈ ∞  2   A∗ f ( X∗ ) AA∗X∗B   | |     2  B∗XAB∗1( X ) B | | is positive. Proof. For any x, y in H 2 1 2 1 B∗XAx, y = XAx, By 6 f ( X∗ ) Ax, Ax 2 1( X ) By, By 2 . |h i| |h i| h | | i h | | i By Lemma 2.1  2   A∗ f ( X∗ ) AA∗X∗B   | |    > 0.  2  B∗XAB∗1( X ) B | | M. Taleb Alfakhr, M. E. Omidvar / Filomat 32:8 (2018), 2861–2866 2863

The third Lemma is Theorem 2.1 in [1]. " # AC∗ Lemma 2.3. Let A, B and C be compact operators in ( ) and is positive. Then sj(C) 6 sj(A B) for B H CB ⊕ j = 1, 2,.... Now we establish a general singular value inequality,from which singular value inequalities for products of operators follow as special cases.

Theorem 2.4. Let Ai,Bi and Xi be operators in ( ), where Xi is compact operator i = 1, 2,..., n and let f and 1 be non-negative functions on [0, ), which are continuousB H and satisfy the relation f (t)1(t) = t for all t [0, ), also let h be a non-negative increasing∞ operator convex function on [0, ). Then ∈ ∞     ∞  Xn Xn Xn     2 2  h sj  ωiA∗X∗Bi sj  ωih(A∗ f ( X∗ ) Ai) ωih(B∗1( Xi ) Bi)   i i  ≤  i | i | ⊕ i | |  i=1 i=1 i=1 Pn for j = 1, 2,... and positive real numbers ωi such that i=1 ωi = 1. Proof. The matrix for i = 1, 2,..., n  2   Ai∗ f ( Xi∗ ) Ai Ai∗Xi∗Bi   | |    > 0 (by Lemma 2.2).  2  B∗XiAi B∗1( Xi ) Bi i i | | So, by Lemma 2.3 and some property of singular values, we have     Xn Xn Xn    2 2  sj  A∗X∗Bi 6 sj  A∗ f ( X∗ ) Ai B∗1( Xi ) Bi . (4)  i i   i | i | ⊕ i | |  i=1 i=1 i=1 for j = 1, 2,.... Now consider the non-negative increasing operator convex function h on [0, ) and in the ∞ inequality (4), put √ωiAi, √ωiBi instead of Ai, Bi respectively . It follows that       Xn Xn Xn      2 2  h sj  ωiA∗X∗Bi 6 h sj  ωiA∗ f ( X∗ ) Ai ωiB∗1( Xi ) Bi   i i    i | i | ⊕ i | |  i=1 i=1 i=1    Xn   2 2  = h sj  ωi(A∗ f ( X∗ ) Ai B∗1( Xi ) Bi)   i | i | ⊕ i | |  i=1    Xn   2 2  = sj h  ωi(A∗ f ( X∗ ) Ai B∗1( Xi ) Bi)   i | i | ⊕ i | |  i=1 (by elementary functional calculus)   Xn  2 2  6 sj  ωih(A∗ f ( X∗ ) Ai B∗1( Xi ) Bi)  i | i | ⊕ i | |  i=1 (h is operator convex)   Xn Xn  2 2  = sj  ωih(A∗ f ( X∗ ) Ai) ωih(B∗1( Xi ) Bi)  i | i | ⊕ i | |  i=1 i=1 for j = 1, 2,....

Corollary 2.5. Let Ai, Bi, Xi ( ) such that Xi is compact for i = 1, 2,..., n. Then ∈ B H  n   n n  X X r X r r    2 2  s  ωiA∗X∗Bi sj  ωi A∗ f ( X∗ ) Ai ωi B∗1( Xi ) Bi  j  i i  ≤  i | i | ⊕ i | |  i=1 i=1 i=1 for j = 1, 2,.... M. Taleb Alfakhr, M. E. Omidvar / Filomat 32:8 (2018), 2861–2866 2864

Proof. Let h(x) = xr (1 r 2) in Theorem 2.4, and we get the result. ≤ ≤ A particular case of Theorem 2.4 and Corollary 2.5 can be seen as follows.

Corollary 2.6. Let Ui, Vi, Xi ( ) such that Ui,Vi are unitary and Xi is compact for i = 1, 2,..., n. Then ∈ B H     Xn Xn Xn r    2r 2r  s  ωiU∗X∗Vi sj  ωiU∗ f ( X∗ ) Ui ωiV∗1( Xi ) Vi j  i i  ≤  i | i | ⊕ i | |  i=1 i=1 i=1 j = 1, 2,.... In particular,     Xn Xn Xn r    r r  s  U∗X∗Vi sj  U∗ X∗ Ui V∗ Xi Vi j  i i  ≤  i | i | ⊕ i | |  i=1 i=1 i=1 j = 1, 2,....

r Proof. The result follows from Theorem 2.4 by letting h(x) = x (1 r 2), Ai = Ui and Bi = Vi unitary ≤ ≤ 1 operators for i = 1, 2,..., n, and using Lemma 1.6 in [6]. The particular case follows by letting f (t) = 1(t) = t 2 1 and ωi = n for i = 1, 2,..., n. As an application of the inequality (4), we get the following corollaries.

Corollary 2.7. Let Ai, Bi, Xi ( ) such that Xi is compact for i = 1, 2,..., n. Then ∈ B H

Xn Xn Xn 2 2 A∗X∗Bi A∗ f ( X∗ ) Ai B∗1( Xi ) Bi (5) i i ≤ i | i | ⊕ i | | i=1 i=1 i=1 for every unitarily invariant norm. In particular,   Xn Xn Xn  2 2  A∗X∗Bi max  A∗ f ( X∗ ) Ai , B∗1( Xi ) Bi  i i ≤  i | i | i | |  i=1 i=1 i=1 and

1  p p p Xn  Xn Xn   2 2  A∗X∗Bi  A∗ f ( X∗ ) Ai + B∗1( Xi ) Bi  i i ≤  i | i | i | |  i=1 p i=1 p i=1 p for 1 p < . ≤ ∞ 1 Remark 2.8. Let A and B be in n(C). Putting n = 2, f (t) = 1(t) = t 2 ,A = A, A = B, B = B, B = A and M 1 2 1 2 X1∗ = X2∗ = I in the inequality (5), we get

(A∗B + B∗A) 0 (A∗A + B∗B) (A∗A + B∗B) ||| ⊕ ||| ≤ ||| ⊕ ||| for every unitarily invariant norm, which was given by Hirzallah and Kittaneh in [8].

Corollary 2.9. Let Xi ( ) be compact for i = 1, 2,..., n. Then ∈ B H

Xn Xn

X∗ 6 ( X∗ Xi ) i | i |⊕| | i=1 i=1 for every unitarily invariant norm.

1 1 2 Proof. Put Ai∗ = Ai = Bi∗ = Bi = I for i = 1, 2,..., n, and f (t) = (t) = t in the inequality (5) and we get the result. M. Taleb Alfakhr, M. E. Omidvar / Filomat 32:8 (2018), 2861–2866 2865

Corollary 2.10. Let Ai, X ( ), i = 1, 2,..., n such that X is a compact positive operator. Then ∈ B H n X 1 1 1 1 2 2 A1XA∗ + A2XA∗ + + AnXA∗ 6 X 2 Ai X 2 X 2 Ai X 2 (6) 2 3 ··· 1 | | ⊕ | | i=1 for every unitarily invariant norm.

Proof. Put A1∗ = A1, B1 = A2∗ , A2∗ = A2, B2 = A3∗ ,..., An∗ = An, Bn = A1∗ , Xi∗ = X for i = 1, 2, ..., n and 1 f (t) = 1(t) = t 2 for all t [0, ) in the inequality (5) and we have ∈ ∞

A1XA2∗ + A2XA3∗ + + AnXA1∗ ··· Xn Xn

6 A∗XAi A∗XAi i ⊕ i i=1 i=1 n " #" #" # X A∗ 0 X 0 Ai 0 6 i 0 A∗ 0 X 0 Ai i=1 i n " # 1 " # " # 1 " # X 2 ∗ 2  X 0 A∗ 0   X 0 A∗ 0  =  i   i   0 X 0 A∗   0 X 0 A∗  i=1 i i n " # 1 " #2 " # 1 X 2 2 X 0 Ai 0 X 0 = | | (since T∗T = TT∗ ) 0 X 0 Ai 0 X || || || || i=1 | | Xn 1 2 1 1 2 1 = X 2 Ai X 2 X 2 Ai X 2 | | ⊕ | | i=1 for every unitarily invariant norm.

Remark 2.11. In n(C) if taken Xi = I, for i = 1, 2,..., n in (6), then M n n X 1 X 2 2 p 2 A1A∗ + A2A∗ + + AnA∗ 6 Ai Ai p= 2 Ai 2 3 ··· 1 p k| | ⊕| | k k kp i=1 i=1 for 1 6 p < , which is another version of the inequality in [5, Corollary 2.3 ]. ∞ Corollary 2.12. Let A, B n(C). Then ∈ M 2 2 2 2 sj(A∗A B∗B) 6 sj A + B A + B − | | | | ⊕ | | | | for j = 1, 2,..., n.

1 Proof. Put n = 2, f (t) = 1(t) = t 2 , A∗ = (A + B)∗, B = (A B), A∗ = (A B)∗, B = (A + B), X∗ = X∗ = I, in the 1 1 − 2 − 2 1 2 inequality (4), also 2(A∗A B∗B) = (A + B)∗(A B) + (A B)∗(A + B). We have − − − 2sj(A∗A B∗B) = sj (A + B)∗ (A B) + (A B)∗ (A + B) − − − 6 sj ((A + B)∗(A + B) + (A B)∗(A B)) − − ((A B)∗(A B) + (A + B)∗(A + B)) ⊕ − − = sj 2(A∗A + B∗B) 2(A∗A + B∗B) ⊕ 2 2 2 2 = 2sj A + B A + B | | | | ⊕ | | | | for j = 1, 2,.... This completes the proof. M. Taleb Alfakhr, M. E. Omidvar / Filomat 32:8 (2018), 2861–2866 2866

Based on Theorem 2.4 and the inequality (1), we have the following singular value of the operator AX∗B. Theorem 2.13. Let A, B, X ( ) such that X is compact, A and B are self-adjoint with A 6 a and B 6 b for some real numbers a and b. Then∈ B H | | | |

2 sj(AX∗B) 6 max a, b sj(X X) (7) { } ⊕ for j = 1, 2,.... Proof. Let " # A 0 C = . |0| B | | Note that C is positive and C 6 max a, b , therefore { } C 6 max a, b . || || { } Hence

sj(AX∗B) 6 sj(A X∗ A B X B) (by Theorem 2.4) | | ⊕ | | 2 6 C sj(X X) (by the inequality (1)) || || ⊕ 2 6 max a, b sj(X X) { } ⊕ for j = 1, 2,....

Corollary 2.14. Let A, B, X, Y ( ) such that X and Y are compact and A, B are as in Theorem 2.13, then ∈ B H 2 sj(AX∗B + AY∗B) 6 max a, b sj((X + Y) (X + Y)). (8) { } ⊕ Proof. The inequality (8) follows from the inequality (7) by replacing the operator X by X + Y.

References

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